OK, it has something to do with the option to "Open a Folder". I can open the file directly into Visual Studio, enter the "Rename" panel (F2), and see the references that will be affected by any name change.
However, if I open the containing folder (CTRL + SHFT + ALT + O), and then open the very same file, the Rename functionality is broken.
I have always worked on this series of files in this same manner (opening the folder), and only recently saw this problematic behavior... so I'm thinking this must have changed with a VS update.